(4f-3g)*(5f+6g)=0

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Solution for (4f-3g)*(5f+6g)=0 equation: 


Simplifying
(4f + -3g)(5f + 6g) = 0

Multiply (4f + -3g) * (5f + 6g)
(4f * (5f + 6g) + -3g * (5f + 6g)) = 0
((5f * 4f + 6g * 4f) + -3g * (5f + 6g)) = 0

Reorder the terms:
((24fg + 20f2) + -3g * (5f + 6g)) = 0
((24fg + 20f2) + -3g * (5f + 6g)) = 0
(24fg + 20f2 + (5f * -3g + 6g * -3g)) = 0
(24fg + 20f2 + (-15fg + -18g2)) = 0

Reorder the terms:
(24fg + -15fg + 20f2 + -18g2) = 0

Combine like terms: 24fg + -15fg = 9fg
(9fg + 20f2 + -18g2) = 0

Solving
9fg + 20f2 + -18g2 = 0

Solving for variable 'f'.

Factor a trinomial.
(4f + -3g)(5f + 6g) = 0


Subproblem 1
Set the factor '(4f + -3g)' equal to zero and attempt to solve:

Simplifying
4f + -3g = 0

Solving
4f + -3g = 0

Move all terms containing f to the left, all other terms to the right.

Add '3g' to each side of the equation.
4f + -3g + 3g = 0 + 3g

Combine like terms: -3g + 3g = 0
4f + 0 = 0 + 3g
4f = 0 + 3g
Remove the zero:
4f = 3g

Divide each side by '4'.
f = 0.75g

Simplifying
f = 0.75g

Subproblem 2
Set the factor '(5f + 6g)' equal to zero and attempt to solve:

Simplifying
5f + 6g = 0

Solving
5f + 6g = 0

Move all terms containing f to the left, all other terms to the right.

Add '-6g' to each side of the equation.
5f + 6g + -6g = 0 + -6g

Combine like terms: 6g + -6g = 0
5f + 0 = 0 + -6g
5f = 0 + -6g
Remove the zero:
5f = -6g

Divide each side by '5'.
f = -1.2g

Simplifying
f = -1.2g
Solution
f = {0.75g, -1.2g}

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